Nonaffine Fluctuations Research Articles

Effective medium theory for mechanical phase transitions of fiber networks.

Networks of stiff fibers govern the elasticity of biological structures such as the extracellular matrix of collagen. These networks are known to stiffen nonlinearly under shear or extensional strain. Recently, it has been shown that such stiffening is governed by a strain-controlled athermal but critical phase transition, from a floppy phase below the critical strain to a rigid phase above the critical strain. While this phase transition has been extensively studied numerically and experimentally, a complete analytical theory for this transition remains elusive. Here, we present an effective medium theory (EMT) for this mechanical phase transition of fiber networks. We extend a previous EMT appropriate for linear elasticity to incorporate nonlinear effects via an anharmonic Hamiltonian. The mean-field predictions of this theory, including the critical exponents, scaling relations and non-affine fluctuations qualitatively agree with previous experimental and numerical results.

Steady state dynamic dependence between local mobility and non-affine fluctuations in two-dimensional aggregates

Motivated by qualitative experimental observations in collective behavior of self-propelled camphor particles at air-water interfaces, we study a generic aggregate forming system in two dimensions using canonical ensemble constant temperature molecular dynamics simulation. The aggregates form due to the competition between short-range attraction and long-range repulsion of pair-wise interactions as a generic proxy for the specific case of short-range capillary attraction competing with long-range Marangoni-assisted repulsion in camphor boat systems. Choosing the appropriate set of interaction parameters, we focus on characterising the local dynamics in two specific limiting morphologies, viz. compact and string-like aggregates. We focus on the temporal evolution of the mobility of an individual particle and the dynamic change in its nearest neighbourhood, measured in terms of the Debye–Waller factor () and the non-affine parameter (), respectively (both defined in the text), and their interrelation over several lengths of observation time . The distribution for both measures are found to follow the relation: for the measured quantity x. The exponent is equal to two and one respectively, for the compact and string-like morphologies following the respective ideal fractal dimension of these aggregates. A functional dependence between these two observables is determined from a detailed statistical analysis of their joint and conditional distributions. The results obtained can readily be used and verified by experiments on aggregate forming systems more generic than the specific camphor boat system that motivated us, such as globular proteins, nanoparticle self-assembly etc. Further, the insights gained from this study might be useful to understand the evolution of collective dynamics in diverse glass-forming systems.

Excess vibrational modes of a crystal in an external non-affine field

Thermal displacement fluctuations in a crystal may be classified as either “affine” or “non-affine”. While the former couples to external stress with familiar consequences, the response of a crystal when non-affine displacements are enhanced using the thermodynamically conjugate field, is relatively less studied. We examine this using a simple model of a crystal in two dimensions for which analytical calculations are possible. Enhancing non-affine fluctuations destabilises the crystal. The population of small frequency phonon modes increases, with the phonon density of states shifting, as a whole, towards zero frequency. Even though the crystal is free of disorder, we observe growing length and time scales. Our results, which may have implications for the glass transition and structural phase transitions in solids, are compared to molecular dynamics simulations. Possibility of experimental verification of these results is also discussed. We show that it is possible to enhance non-affine displacement fluctuations in a crystal using an external field. Although transient, these fluctuations can shift the phonon density of states to low frequencies and cause “Boson peak” like phenomenon in an otherwise perfect crystal without quenched disorder.

Contrasting the dynamics of elastic and non-elastic deformations across an experimental colloidal Martensitic transition.

We present a framework to segregate the roles of elastic and non-elastic deformations in the examination of real-space experiments of solid-solid Martensitic transitions. The Martensitic transformation of a body-centred-tetragonal (BCT) to a body-centred-orthorhombic (BCO) crystal structure has been studied in a model system of micron-scale ionic microgel colloids (P. S. Mohanty, P. Bagheri, S. Nöjd, A. Yethiraj and P. Schurtenberger, Phys. Rev. X, 2015, 5, 011030). Non-affine fluctuations, i.e., displacement fluctuations that do not arise from purely elastic (affine) deformations, are detected in particle configurations acquired from the experiment. Tracking these fluctuations serves as a highly sensitive tool in signaling the onset of the Martensitic transition and precisely locating particle rearrangements occurring at length scales of a few particle diameters. Particle rearrangements associated with non-affine displacement modes become increasingly favorable during the transformation process. The nature of the displacement fluctuation modes that govern the transformation are shown to be different from those predominant in an equilibrium crystal. We show that BCO crystallites formed through shear may, remarkably, co-exist with those resulting from local rearrangements within the same sample.

Slow creep in soft granular packings.

Transient creep mechanisms in soft granular packings are studied numerically using a constant pressure and constant stress simulation method. Rapid compression followed by slow dilation is predicted on the basis of a logarithmic creep phenomenon. Characteristic scales of creep strain and time exhibit a power-law dependence on jamming pressure, and they diverge at the jamming point. Microscopic analysis indicates the existence of a correlation between rheology and nonaffine fluctuations. Localized regions of large strain appear during creep and grow in magnitude and size at short times. At long times, the spatial structure of highly correlated local deformation becomes time-invariant. Finally, a microscale connection between local rheology and local fluctuations is demonstrated in the form of a linear scaling between granular fluidity and nonaffine velocity.

Pre-yield non-affine fluctuations and a hidden critical point in strained crystals.

A crystalline solid exhibits thermally induced localised non-affine droplets in the absence of external stress. Here we show that upon an imposed shear, the size of these droplets grow until they percolate at a critical strain, well below the value at which the solid begins to yield. This critical point does not manifest in most thermodynamic or mechanical properties, but is hidden and reveals itself in the onset of inhomogeneities in elastic moduli, marked changes in the appearance and local properties of non-affine droplets and a sudden enhancement in defect pair concentration. Slow relaxation of stress and an-elasticity appear as observable dynamical consequences of this hidden criticality. Our results may be directly verified in colloidal crystals with video microscopy techniques but are expected to have more general validity.

Non-affine fluctuations and the statistics of defect precursors in the planar honeycomb lattice

Certain localised displacement fluctuations in the planar honeycomb lattice may be identified as precursors to topological defects. We show that these fluctuations are among the most pronounced non-affine distortions of an elemental coarse graining volume of the honeycomb structure at non zero temperatures. We obtain the statistics of these precursor modes in the canonical ensemble, evaluating exactly their single point and two-point spatio-temporal distributions, for a lattice with harmonic nearest neighbour and next near neighbour bonds. As the solid is destabilised by tuning interactions, the precursor fluctuations diverge and correlations become long-lived and long-ranged.

Statistics of non-affine defect precursors: tailoring defect densities in colloidal crystals using external fields.

Coarse-graining atomic displacements in a solid produces both local affine strains and "non-affine" fluctuations. Here we study the equilibrium dynamics of these coarse grained quantities to obtain space-time dependent correlation functions. We show how a subset of these thermally excited, non-affine fluctuations act as precursors for the nucleation of lattice defects and suggest how defect probabilities may be altered by an experimentally realisable "external" field conjugate to the global non-affinity parameter. Our results are amenable to verification in experiments on colloidal crystals using commonly available holographic laser tweezer and video microscopy techniques, and may lead to simple ways of controlling the defect density of a colloidal solid.

Shear shocks in fragile networks.

A minimal model for studying the mechanical properties of amorphous solids is a disordered network of point masses connected by unbreakable springs. At a critical value of its mean connectivity, such a network becomes fragile: it undergoes a rigidity transition signaled by a vanishing shear modulus and transverse sound speed. We investigate analytically and numerically the linear and nonlinear visco-elastic response of these fragile solids by probing how shear fronts propagate through them. Our approach, which we tentatively label shear front rheology, provides an alternative route to standard oscillatory rheology. In the linear regime, we observe at late times a diffusive broadening of the fronts controlled by an effective shear viscosity that diverges at the critical point. No matter how small the microscopic coefficient of dissipation, strongly disordered networks behave as if they were overdamped because energy is irreversibly leaked into diverging nonaffine fluctuations. Close to the transition, the regime of linear response becomes vanishingly small: the tiniest shear strains generate strongly nonlinear shear shock waves qualitatively different from their compressional counterparts in granular media. The inherent nonlinearities trigger an energy cascade from low to high frequency components that keep the network away from attaining the quasi-static limit. This mechanism, reminiscent of acoustic turbulence, causes a superdiffusive broadening of the shock width.

Nonlinear effective-medium theory of disordered spring networks

Disordered soft materials, such as fibrous networks in biological contexts, exhibit a nonlinear elastic response. We study such nonlinear behavior with a minimal model for networks on lattice geometries with simple Hookian elements with disordered spring constant. By developing a mean-field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations, we provide insight into the origins of the strain stiffening and softening behavior of these systems. We find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity. In particular, the nonlinear response is controlled by the isostatic connectivity, which depends strongly on the applied strain. Our predictions for the strain dependence of the isostatic point as well as the strain-dependent differential bulk modulus agree well with numerical results in both two and three dimensions. In addition, by using a mapping between the disordered network and a regular network with random forces, we calculate the nonaffine fluctuations of the deformation field and compare them to the numerical results. Finally, we discuss the limitations and implications of the developed theory.

Nonaffine Transient Network Theory of Associating Polymer Solutions

We studied the effect of nonaffine displacement of micellar junctions on the linear viscoelasticity of transient networks formed with telechelic associating polymers. For a junction of small aggregation number, the recoil motion is large when a hydrophobic end chain dissociates from it and, hence, leads to large nonaffine fluctuations. We show that diffusive motion around the mean position of the junction gives rise to nonsymmetric deviation of the loss modulus from that of Maxwell fluid with a single relaxation time. We calculate the high-frequency shoulder and the degree of low-frequency softening in the loss modulus as functions of the parameter Dβ0-1/na2, the mean radius of fluctuations during the end-chain lifetime divided by the mean-square end-to-end distance of a bridge chain. The degree of softening in the storage modulus at low frequencies is also calculated. We find that nonlinearity in the tension-elongation curve of a chain significantly affects the linear viscoelasticity due to fluctuations of the end-to-end distance.

Neutron scattering from strained polymer networks

The scattering function S(k) for neutrons off a randomly cross-linked phantom network is calculated. Adopting the results of Deam and Edwards (1976) in which the mean position of the chain deforms affinely and fluctuations about this mean are random (i.e. non-affine) gives very different results from Benoit (1975) and co-workers who assume affine deformation of the chain shape. Some of the results obtained are similar to those of Pearson (1977) who calculates S(k) using Flory's theory (1976) of non-affine fluctuations. However, the predictions for networks more densely cross-linked are different from those of Flory's theory.